Semi-equivelar maps on the torus Dipendu Maity Department of - PowerPoint PPT Presentation

Semi-equivelar maps on the torus Dipendu Maity Department of Mathematics, Indian Institute of Science, Bangalore, India October 3, 2016 Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 1 / 18

Objective • What types of maps exist on the torus? • What is the status of the classification of these maps? • Are all these maps vertex-transitive? Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 2 / 18

Eembedding and Map Graph embedding on surfaces : An embedding of a graph G in a surface S is an one-one mapping i : G → S . • A map M is an embedding of a connected finite simple graph G into a surface S in which closure of a connected components of S \ G is homeomorphic to closed 2-disk. • The components are called faces of M and each face is a n -gon ( n ≥ 3). So, M =: ( V , E , F ). v 3 v 4 v 5 v 6 v 7 v 1 v 2 v 3 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 Map on the torus Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 3 / 18

Regular, Semiregular, Equivelar and Semi-equivelar map Regular map : A map is said to be regular if the symmetry group of the map acts transitively on the flags (an incident vertex-edge-face triple) of the map. Semiregular map : A map is said to be semiregular if the symmetry group of the map acts transitively on the set of incident vertex-edge pairs of the map. Equivelar map : A map M in which each face of M is a p -gon and each vertex belongs to exactly q faces is called { p q } -equivelar map. Semi-equivelar map : A map M is said to be a semi-equivelar map of type { a p , b q , . . . , m r } if face sequence of each vertex of M is { a p , b q , . . . , m r } . • Class of regular map ⊂ Class of Equivelar map. • Class of semiregular map ⊂ Class of Semi-Equivelar map. Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 4 / 18

Regular and Semi-regular maps on the surfaces of χ = 0 Proposition : If { p n 1 1 , . . . , p n k k } satisfies any of the following two properties then { p n 1 1 , . . . , p n k k } can not be the type of any semi-equivelar map on a surface. There exists i such that n i = 1 , p i is odd, p j � = p i for all j � = i and 1 p i − 1 � = p i +1 . There exists i such that n i = 2, p i is odd and p j � = p i for all j � = i . 2 Corollary : Let X be a semi-equivelar map on a surface M . If M is the torus or the Klein bottle then the type of X is { 3 6 } , { 4 4 } , { 6 3 } , { 3 3 , 4 2 } , { 3 2 , 4 , 3 , 4 } , { 3 , 6 , 3 , 6 } , { 3 4 , 6 } , { 4 , 8 2 } , { 3 , 12 2 } , { 4 , 6 , 12 } , { 3 , 4 , 6 , 4 } . • The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus. Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 5 / 18

Semi-equivelar maps on torus for vertices ≤ 15 • Among 11 types { 3 6 } , { 4 4 } , { 6 3 } are studied on the torus by Altshuler [1], Brehm and K¨ uhnel [2], Kurth [6], Negami [8], Datta and Upadhyay [4]. • In 2015, the classification of remaining eight types { 3 3 , 4 2 } , { 3 2 , 4 , 3 , 4 } , { 3 , 6 , 3 , 6 } , { 3 4 , 6 } , { 4 , 8 2 } , { 3 , 12 2 } , { 4 , 6 , 12 } , { 3 , 4 , 6 , 4 } are being attempted by Tiwari and Upadhyay [9]. Proposition : In [9], there are exactly 11 semi-equivelar maps of types { 3 3 , 4 2 } , { 3 2 , 4 , 3 , 4 } , { 3 , 6 , 3 , 6 } , { 3 4 , 6 } , { 4 , 8 2 } , { 3 , 12 2 } , { 4 , 6 , 12 } , { 3 , 4 , 6 , 4 } with ≤ 15 vertices on the surfaces of Euler characteristic 0. Six of these maps are orientable and five are non-orientable. Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 6 / 18

Semi-equivelar maps on the torus : maps of type { 3 3 , 4 2 } Let M be a map of type { 3 3 , 4 2 } on the torus. We define three paths in M through each vertex. Definition 1 : Let P 1 := P ( . . . , u i − 1 , u i , u i +1 , . . . ) be a path in edge graph of M . We say P 1 of type A 1 if all the triangles incident with an inner (degree two in P 1 ) vertex u i lie on one side and all quadrangles incident with u i lie on the other side of the portion of P 1 for all i . Definition 2 : Let P 2 := P ( . . . , v i − 1 , v i , v i +1 , . . . ) be a path in edge graph of M for which v i , v i +1 be two consecutive inner vertices of P 2 or an extended path of P 2 . Then we say P 2 of type A 2 if lk ( v i ) = C ( a , v i − 1 , b , c , v i +1 , d , e ) implies lk ( v i +1 ) = C ( a 0 , v i +2 , b 0 , d , v i , c , p ) and lk ( v i ) = C ( x , v i +1 , z , l , v i − 1 , k , m ) implies lk ( v i +1 ) = C ( l , v i , m , x , v i +2 , g , z ). Definition 3 : Let P 3 := P ( . . . , w i − 1 , w i , w i +1 , . . . ) be a path in edge graph of M for which w i , w i +1 are two inner vertices of P 3 or an extended path of P 3 . Then we say P 3 of type A 3 if lk ( w i ) = C ( a , w i − 1 , b , c , d , w i +1 , e ) implies lk ( w i +1 ) = C ( a 1 , w i +2 , b 1 , p , e , w i , d ) and lk ( w i ) = C ( a 2 , w i +1 , b 2 , p , e , w i − 1 , d ) implies lk ( w i +1 ) = C ( p , w i , d , a 2 , z 1 , w i +2 , b 2 ). Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 7 / 18

A 2 A 3 A 1 v Paths of types A 1 , A 2 , A 3 through vertex v Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 8 / 18

• Let Q be a maximal path of type A t for some t ∈ { 1 , 2 , 3 } . Then, there exists an edge e such that Q ∪ e is a cycle of type A t and non-contractible. u r − 6 w r − 10 w i +3 u i +7 u r − 5 w r − 9 w i +2 u i +6 u r − 4 w r − 8 w i +1 u i +5 u r − 3 w i u i +4 u r − 2 w i − 1 u i +3 u r − 1 u i +2 u r u i +1 u i u i − 1 Cycle of type A 2 Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 9 / 18

• The map M has a T ( r , s , k ) planar polyhedral representation. . . . . . . v k +1 v k +2 v k +3 v k +4 v n v 1 v 2 v 3 v k − 2 v k − 1 v k v k +1 . . . . . . z 1 z 2 z 3 z 4 z k z k +1 z k +2 z k +3 z r − 2 z r − 1 z r z 1 . . . . . . x 1 x 2 x 3 x 4 x k x k +1 x k +2 x k +3 x r − 2 x r − 1 x r x 1 . . . . . . w 1 w 2 w 3 w 4 w k w k +1 w k +2 w k +3 w r − 2 w r − 1 w r w 1 . . . . . . v 1 v 2 v 3 v 4 v k v k +1 v k +2 v k +3 v r − 2 v r − 1 v r v 1 Figure : T ( r , 4 , k ) • Let M be a map of type { 3 3 , 4 2 } on the torus. Then, the cycles of type A 1 have unique length and the cycles of type A 2 have at most two different lengths. • The { 3 3 , 4 2 } -maps of the form T ( r , s , k ) exist if and only if the following holds : (i) s ≥ 2 even, (ii) r ≥ 3, (iii) n = rs ≥ 10, (iv) 2 ≤ k ≤ r − 3 if s = 2 & 0 ≤ k ≤ r − 1 if s ≥ 4. Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 10 / 18

v 4 v 5 v 6 v 7 v 1 v 2 v 3 v 4 v 4 v 5 v 6 v 7 v 1 v 2 v 3 v 4 z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 1 z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 1 w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 1 w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 1 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 Figure 2 : T (7 , 4 , 3) Figure 1 : T (7 , 4 , 3) • Let T i be a map on n i vertices and n 1 = n 2 . Then, T 1 ∼ = T 2 if and only if they have cycles of types A 1 , A 2 , A 3 , A 4 of same length. u 3 u 4 u 5 u 6 u 7 u 1 u 2 u 3 v 5 v 6 v 7 v 1 v 2 v 3 v 4 v 5 v 8 v 9 v 10 v 11 v 12 v 13 v 14 v 8 u 8 u 9 u 10 u 11 u 12 u 13 u 14 u 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 1 Figure 3 : T(7, 2, 2) : O 1 Figure 4: T(7, 2, 4) : O 2 v 3 v 2 v 1 v 7 v 6 v 5 v 4 v 3 w 4 w 5 w 6 w 7 w 1 w 2 w 3 w 4 v 12 v 11 v 10 v 9 v 8 v 14 v 13 v 12 w 8 w 9 w 10 w 11 w 12 w 13 w 14 w 8 v 5 v 4 v 3 v 2 v 1 v 7 v 6 v 5 w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 1 Figure 6 : T(7 , 2 , 2) Figure 5 : T(7, 2, 3) : O 3 Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 11 / 18

• Let M i be a map on n i vertices. Let T i = T ( r i , s i , k i ) denote a ( r i , s i , k i ) - representation of M i for i ∈ { 1 , 2 } . Then, T 1 �∼ = T 2 ∀ r 1 � = r 2 , T 1 �∼ = T 2 ∀ s 1 � = s 2 , T ( r 1 , s 1 , k 1 ) �∼ = T ( r 1 , s 1 , k 2 ) for s 1 = 2 and k 2 ∈ { 2, 3, . . . , r 1 − 3 } \ { k 1 , r 1 − k 1 − 1 } , T ( r 1 , s 1 , k 1 ) �∼ = T ( r 1 , s 1 , k 2 ) for s 1 ≥ 4 2 } , T ( r 1 , s 1 , k 1 ) ∼ and k 2 ∈ { 0, 1, . . . , r 1 − 1 } \ { k 1 , r 1 − k 1 − s 1 = T ( r 1 , s 1 , r 1 − k 1 − 1) for s 1 = 2 and r 1 ≥ 5, and T ( r 1 , s 1 , k 1 ) ∼ = T ( r 1 , s 1 , r 1 − s 1 2 − k 1 ) for s 1 ≥ 4 and r 1 ≥ 3. Table : Maps of type { 3 3 , 4 2 } Equivalence classes Length of cycles i ( n ) n 10 T(5, 2, 2) (5, { 10 , 10 } , 4) 1(10) 12 T(6, 2, 2), T(6, 2, 3) (6, { 6 , 4 } , 4) 3(12) T(3, 4, 0), T(3, 4, 1) (3, { 4 , 12 } , 4) T(3, 4, 2) (3, { 12 , 12 } , 6) 14 T(7, 2, 2), T(7, 2, 4) (7, { 14 , 14 } , 4) 2(14) T(7, 2, 3) (7, { 14, 14 } , 5) 16 T(8, 2, 2), T(8, 2, 5) (8, { 8 , 16 } , 4) 5(16) T(8, 2, 3), T(8, 2, 4) (8, { 16 , 4 } , 5) T(4, 4, 0), T(4, 4, 2) (4, { 4 , 8 } , 4) T(4, 4, 1) (4, { 16 , 16 } , 5) T(4, 4, 3) (4, { 16 , 16 } , 7) 18 T(9, 2, 2), T(9, 2, 6) (9, { 18 , 6 } , 4) 5(18) T(9, 2, 3), T(9, 2, 5) (9, { 6 , 18 } , 5) T(9, 2, 4) (9, { 18 , 18 } , 6) T(3, 6, 0) (3, { 6 , 6 } , 6) T(3, 6, 1), T(3, 6, 2) (3, { 18 , 18 } , 7) Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 12 / 18